CN111178485A - Multi-target evolutionary algorithm based on double population cooperation - Google Patents

Multi-target evolutionary algorithm based on double population cooperation Download PDF

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CN111178485A
CN111178485A CN201910838388.5A CN201910838388A CN111178485A CN 111178485 A CN111178485 A CN 111178485A CN 201910838388 A CN201910838388 A CN 201910838388A CN 111178485 A CN111178485 A CN 111178485A
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谢承旺
张飞龙
周慧
闭应洲
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Nanning Normal University
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Abstract

The invention provides a multi-target evolutionary algorithm based on double-population cooperation, which adopts a randomization method and a mixed level orthogonal initialization method to respectively generate two initial sub-populations. The simulated binary crossover operation and the differential mutation operation are alternately applied to the two sub-populations in generation-by-generation evolution to generate offspring individuals. The two sub-populations and their offspring individuals are merged separately to generate two temporary intermediate populations, which are then subjected to a fast non-dominant ranking operation to select the superior individual from the two intermediate populations to update the external profile set. In the whole evolution process, the two sub-populations keep respective evolution modes, and meanwhile, the two sub-populations can share and interact information through an external archive set. The invention carries out organic cooperation on the strategies and the methods, better balances the global exploration capacity and the local exploitation capacity of the multi-target evolutionary algorithm, and effectively solves the problems that the multi-target evolutionary algorithm is easy to premature convergence, the diversity of the obtained solution set is not good, and the like.

Description

Multi-target evolutionary algorithm based on double population cooperation
Technical Field
The invention relates to an intelligent optimization algorithm, in particular to a multi-objective evolutionary algorithm based on double-population cooperation.
Background
There are a number of problems in scientific computing and engineering practice that require the simultaneous optimization of two or more objectives, commonly referred to as multi-objective optimization problems (MOP). The targets of the MOP problem tend to conflict identically, i.e., improving one of the target values generally causes the other target values to deteriorate. Thus, the MOP problem generally does not have a unique optimal solution so that each target simultaneously obtains an optimal value, but is often a compromise solution, i.e., a Pareto solution set. The Evolutionary Algorithm (EA) is a simple and effective method for solving the MOP problem, the EA can obtain a group of better approximate solutions by running once based on the characteristics of group search, and the EA algorithm does not make special assumption on the mathematical property of the problem to be solved, so that the method is suitable for solving some non-microminiature, non-convex, multi-modal and other complex MOP problems.
To date, many classical multi-objective evolutionary algorithms (MOEA) have emerged, which also exhibit good properties in solving MOP problems, but it must be pointed out that existing MOEA algorithms still exist where improvements are needed, such as: 1) the algorithm is easy to fall into a local optimal area, the global exploration capability and the local exploitation capability are not balanced uniformly, and the diversity and the distribution of the acquired solution set are not good; 2) the existing research of the MOEA algorithm mainly focuses on how to obtain the optimal approximate Pareto frontier, but neglects the synergistic effect among population individuals and between the algorithm and various strategies, and the synergistic evolution idea based on the natural biological population evolution theory is more beneficial to improving the optimization effect of the algorithm, so that the overall performance of the algorithm is improved. Based on the method, the multi-target evolutionary algorithm based on double-population cooperation is designed, not only are competition and cooperation among the sub-populations considered, but also cooperation among the cross strategy and the variation strategy is utilized, so that the overall performance of the algorithm for solving the complex MOP problem is improved.
Without loss of generality, a mathematical model of a minimized multiobjective optimization problem with n decision variables and M objectives can be described as follows:
Figure BDA0002192911830000011
wherein,
Figure RE-GDA0002450258610000026
is a decision vector of n dimensions, and X is a decision space of n dimensions, also called a search space;
Figure RE-GDA0002450258610000027
is an M-dimensional objective function value vector, Y is an objective space with dimension M; f (X) is a function mapped from X to Y, gi(x) Is the ith inequality constraint; q is the number of inequality constraints; h isj(x) Is the jth equality constraint; p is the number of equality constraints; x is the number ofi,minAnd xi,maxAs a decision variable xiLower and upper bounds. The constraint functions g (x) and h (x) together determine the feasible domain of the decision vector x.
Define 1-feasible solution set XfSet of decision vectors, i.e. X, to satisfy constraint functions g (X) and h (X) in equation (1)fX ∈ X | g (X) ≦ 0 and h (X) ═ 0 };
definition of 2-Pareto Branch set x1,x2∈XfIs any two feasible solutions to the multi-objective optimization problem defined in equation (1), called x1Pareto dominate x2(is described as
Figure BDA0002192911830000023
) If and only if
Figure BDA0002192911830000024
Figure BDA0002192911830000025
If true;
definition of 3-Pareto non-dominated solution x*∈XfAnd there are no other solutions
Figure BDA0002192911830000026
So that
Figure BDA0002192911830000027
If it is true and at least one is strictly inequality, then x is called*Is a Pareto non-dominated solution of formula (1), also known as a Pareto optimal solution;
define 4-Pareto optimal solution Set the Pareto optimal solution Set (PS) is the Set of all Pareto optimal solutions in define 3, i.e. PS ═ x*}；
Defining 5-Pareto Frontier (PF) is the projection of the Pareto optimal solution set in definition 4 in the target space, i.e. PF { F (x) | x ∈ PS };
definition of 6-non-dominant solution set of population P (t) is the t generation population of MOEA algorithm, and individuals x*E P (t) is the non-dominant solution of the population, if and only if
Figure BDA0002192911830000028
All non-dominant solutions x in*The set of constituents is referred to as the non-dominant solution set for the population P (t).
Disclosure of Invention
The invention aims to provide a multi-target evolutionary algorithm based on double-population cooperation, which can effectively solve the problems that an MOEA algorithm is easy to fall into local optimization, and the diversity and the distribution of solution sets obtained by the algorithm are poor.
To achieve these objects and other advantages in accordance with the purpose of the invention, there is provided a multi-objective evolutionary algorithm based on dual population cooperation, comprising the steps of:
s1: setting a target number M and a maximum iteration number Tmaxdecision vector dimension N, population scale N, scale of external profile set N', pseudo-binary cross-distribution index η, variation probability pmDifferential evolution scaling factor F;
s2: making iteration counter t equal to 0, utilizing random initialization method to produce initial sub population P whose size is N/2 in search space1(0) Generating an initial sub-population P of size N/2 in a search space by using a mixed horizontal orthogonal initialization method2(0)；
S3: computing a sub-population P1(t) and P2(t) objective function value vectors for each individual;
s4: using a fast non-dominated sorting method to respectively sub-population P according to the objective function value vector of each individual calculated in step S31(t) and P2(t) sorting, copying the non-dominant individuals in the two sub-populations to an external archive set according to a sorting result; if the external archive set is full, executing a diversity maintenance strategy to maintain the external archive set;
s5: pair sub-population P1(t) performing the simulated binary crossover operation to generate the offspring population P thereof1' (t) for the subgroup P2(t) performing selective differential variation to generate a progeny population P thereof2'(t)；
S6: let T1(t)＝P1(t)∪P1'(t)，T2(t)＝P2(t)∪P2' (T), here T1(T) and T2(t) is a temporary intermediate population;
s7: calculating T1(T) and T2(t) objective function value vectors for each individual;
s8: the vectors of the individual objective function values calculated in step S7 are respectively used for T1(T) and T2(T) performing a fast non-dominated sorting, using T according to the sorting result1(t) the first N/2 superior individuals in the population update sub-population P1(T) use of T2(t) the first N/2 superior individuals in the population update sub-population P2(t)；
S9: by P1(t) and P2(t) the non-dominated individual updates the external archive set, and if the external archive set is full, a plurality of maintenance strategies are executed to maintain the external archive set;
s10: two sub-populations P1(t) and P2(t) by substitution, i.e. with P1(t) all individuals update P2(t) with P2All individuals in (t) update P1(t)；
S11: updating an iteration counter: t is t + 1;
s12: judging whether T reaches TmaxIf not, go to step S3, otherwise, go to step S13;
s13: and outputting all the individuals in the external file set, and finishing the algorithm.
Preferably, in the double-population cooperation-based multi-objective evolutionary algorithm S2, an initial sub-population P1(0) And P2(0) Can be respectively expressed as
Figure RE-GDA0002450258610000031
Wherein,
Figure RE-GDA0002450258610000032
Figure RE-GDA0002450258610000033
and t is the iteration number.
Preferably, in the double-population cooperation-based multi-target evolution algorithm S5, the sub-population P1(t) performing the simulated binary crossing to generate its descendants, comprising: from P1(t) randomly selecting two parents
Figure BDA0002192911830000041
And
Figure BDA0002192911830000042
wherein
Figure BDA0002192911830000043
Figure BDA0002192911830000044
To pair
Figure BDA0002192911830000045
And
Figure BDA0002192911830000046
executing the simulated binary cross operation to generate two offspring individuals
Figure BDA0002192911830000047
And
Figure BDA0002192911830000048
wherein
Figure BDA0002192911830000049
While
Figure BDA00021929118300000410
Wherein i is 1,2, … n;
Figure BDA00021929118300000411
is [0,1]]uniformly distributed random numbers are obeyed, and eta is a cross distribution index.
Preferably, in the double-population cooperation-based multi-target evolution algorithm S5, the sub-population P2(t) performing a selected differential mutation operation to generate offspring individuals according to the probability of mutation pmPair sub-population P2(t) the individuals performing selective differential mutation, comprising: from P2(t) selecting an appropriate parent
Figure BDA00021929118300000412
According to
Figure BDA00021929118300000413
The value of the index i selects one of the following three differential variation modes to perform variation operation so as to generate offspring individuals
Figure BDA00021929118300000414
Figure BDA00021929118300000415
Where i is 1,2, …, N/2, t is the number of iterations, F is the scaling factor,
Figure BDA00021929118300000416
to be slave population P2Of the different individuals selected at random,
Figure BDA00021929118300000417
it is a randomly selected non-dominant individual from the external archive set.
Preferably, in the dual-population cooperation-based multi-objective evolutionary algorithm S4 and S9, executing a diversity retention policy on an external archive set is involved, including:
(1) setting an initial value of an epsilon parameter, and setting the scale of an external file set as N';
(2) if the number of individuals in the external file set is less than N', reducing the value of epsilon by the amplitude of the step length gamma generation by generation until the value of epsilon is kept unchanged when the external file set is full;
(3) if a new individual is to be added into the external archive set, firstly, judging an epsilon-Pareto domination relation between the new individual and each individual in the archive set; if the new individual is not dominated by any individual epsilon-Pareto in the external archive set and the external archive set is not full, adding the new individual to the archive, then detecting epsilon-Pareto domination relations among all individuals in the external archive set, and if dominated individuals appear, discarding the dominated individuals; if the new individual and all individuals in the file are in an epsilon-Pareto non-dominated relationship and the file is full, calculating epsilon-congestion distances of the new individual and all individuals in the file set, and discarding the individual with the minimum congestion distance; if the new individual is dominated by the individual epsilon-Pareto in the archive set, the new individual is discarded.
The invention at least comprises the following beneficial effects:
the invention provides a double-population cooperative multi-target evolutionary algorithm, which aims at the problems that the existing multi-target evolutionary algorithm is easy to fall into premature convergence, and the diversity and the distribution of the obtained solution sets are not good. The algorithm firstly adopts two different initialization methods, namely random initialization and mixed horizontal orthogonal initialization, to generate two sub-populations with equal scale. During the generation-by-generation evolution, the two sub-populations are alternately subjected to pseudo-binary crossing and selective differential mutation operations. The algorithm combines the sub-populations and their offspring individuals respectively, and screens out the better individuals in the two combined intermediate populations by using a rapid non-dominated sorting method to update the external archive set. The external archive sets are updated with a coordinated epsilon-congestion distance policy to maintain diversity of solution sets. In the whole evolution process, the two sub-populations keep respective evolution modes, and meanwhile, the two sub-populations can share and interact information through an external archive set. The invention carries out organic cooperation on the strategies, better balances the global exploration capacity and the local exploitation capacity of the algorithm, and effectively solves the problems that the multi-target evolutionary algorithm is easy to premature convergence, the diversity of the obtained solution sets is not good, and the like.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention.
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FIG. 1 is a schematic flow diagram of the present invention;
FIG. 2 is an approximate Pareto front on the test function ZDT1 for the algorithm of the present invention;
FIG. 3 is an approximate Pareto front on the test function DTLZ6 according to the present invention;
FIG. 4 is an approximate Pareto front on test function WFG2 according to the present invention;
fig. 5 is an approximate Pareto front on test function UF4 of the present invention.
Detailed Description
The drawings are merely illustrative and should not be construed as limiting the patent. For a better understanding of the present embodiments, certain features of the drawings may be omitted, enlarged or reduced and do not represent a functional benefit of the actual invention. Certain well-known results in the drawings and omissions thereof will be apparent to those skilled in the art.
The technical solution of the present invention is further described in detail with reference to the accompanying drawings and specific embodiments.
As shown in fig. 1, a multi-objective evolutionary algorithm based on dual population cooperation includes the following processes:
s1: setting a target number M and a maximum iteration number Tmaxdecision vector dimension N, population scale N, scale of external profile set N', pseudo-binary cross-distribution index η, variation probability pmDifferential evolution scaling factor F;
s2: making iteration counter t equal to 0, utilizing random initialization method to produce initial sub population P whose size is N/2 in search space1(0) Generating the scale of N/2 in the search space by using a hybrid horizontal orthogonal initialization methodInitial sub-population P of2(0)；
S3: computing a sub-population P1(t) and P2(t) objective function value vectors for each individual;
s4: using a fast non-dominated sorting method to respectively sub-population P according to the objective function value vector of each individual calculated in step S31(t) and P2(t) sorting, and copying the non-dominant individuals in the two sub-populations to an external archive set according to a sorting result. If the external archive set is full, executing a diversity maintenance strategy to maintain the external archive set;
s5: pair sub-population P1(t) performing the simulated binary crossover operation to generate the offspring population P thereof1' (t) for the subgroup P2(t) performing selective differential variation to generate a progeny population P thereof2'(t)；
S6: let T1(t)＝P1(t)∪P1'(t)，T2(t)＝P2(t)∪P2' (T), here T1(T) and T2(t) is a temporary intermediate population;
s7: calculating T1(T) and T2(t) objective function value vectors for each individual;
s8: respectively aiming at T according to the individual objective function value vectors calculated in S71(T) and T2(T) performing a fast non-dominated sorting, using T according to the sorting result1(t) the first N/2 superior individuals in the population update sub-population P1(t); likewise, using T based on the ranking results2(t) the first N/2 superior individuals in the population update sub-population P2(t)；
S9: by P1(t) and P2The non-dominant individual in (t) updates the external profile set. If the external archive set is full, executing a plurality of maintenance strategies to maintain the external archive;
s10: exchanging two sub-populations P1(t) and P2All of (t);
s11: updating an iteration counter: t is t + 1;
s12: judging whether T reaches TmaxIf not, go to step S3, otherwise,go to step S13;
s13: and outputting all the individuals in the external file set, and finishing the algorithm.
In the embodiment, three classical multi-target evolutionary algorithms, namely NSGA-II, SPEA2 and MOEA/D, are compared with the method of the invention through a simulation experiment, and 5 ZDT series test functions, 7 DTLZ series test functions, 9 WFG series test functions and 7 UF series test functions are selected, and 28 reference multi-target test problems are counted to explain the implementation steps of the invention and verify the effectiveness of the method of the invention. The specific solving steps are as follows:
step 1: the target number M of the ZDT series test functions is 2, where the dimension n of the decision variables of ZDT1, ZDT2 and ZDT3 is 30 and the dimension n of the decision variables of ZDT4 and ZDT6 is 10. The size of the population N is 100, the size of the external archive set N is 100, and the maximum iteration number Tmax250; the target number M of the DTLZ series test function is 3, the dimension of the decision vector is 10, the size of the population is 200, the size of the external file set is 500, and the maximum iteration number T ismax500; the target number M of the WFG series test functions is 2, the dimension of the decision vector is 10, the size N of the population is 200, the size N' of the external file set is 500, and the maximum iteration number Tmax500; the target number M of UF series test functions is 2, the dimension N of decision vector is 30, the size N of population is 300, the size N' of external archive set is 500, and the maximum iteration number Tmax＝1000。
Step 2: making iteration counter t equal to 0, utilizing random initialization method to produce initial sub population P whose size is N/2 in search space1(0) Generating initial sub-population P with size N/2 in search space by mixed horizontal orthogonal initialization method2(0) Here, the initial sub-population P1(0) And P2(0) Can be respectively expressed as
Figure RE-GDA0002450258610000071
Figure RE-GDA0002450258610000072
Wherein,
Figure RE-GDA0002450258610000073
Figure RE-GDA0002450258610000074
and t is the iteration number.
And step 3: computing a sub-population P1(t) and P2(t) objective function value vectors for individual volumes.
And 4, step 4: respectively using a fast non-dominated sorting method to sub-populations P according to the objective function value vector of each individual calculated in the step 31(t) and P2(t) sorting, and copying the non-dominant individuals in the two sub-populations to an external archive set according to a sorting result. And if the external archive set is full, executing a diversity maintenance strategy to maintain the external archive set.
And 5: pair sub-population P1(t) performing the simulated binary crossover operation to generate the offspring population P thereof1' (t) for the subgroup P2(t) performing selective differential variation to generate a progeny population P thereof2'(t)。
Further, the sub-population P1(t) performing the simulated binary crossing to generate its descendants, including, from P1(t) randomly selecting two parents
Figure BDA0002192911830000074
And
Figure BDA0002192911830000075
wherein
Figure BDA0002192911830000076
To pair
Figure BDA0002192911830000077
And
Figure BDA0002192911830000078
executing the simulated binary cross operation to generate two offspring individuals
Figure BDA0002192911830000079
And
Figure BDA00021929118300000710
wherein
Figure BDA00021929118300000711
Figure BDA00021929118300000712
While
Figure BDA00021929118300000713
Wherein i in formula (2) is 1,2, … n,
Figure BDA00021929118300000714
u is [0,1]]and η is a cross distribution index, wherein η is 20 in the embodiment.
Further, the sub-population P2(t) performing a selective differential mutation operation to generate offspring individuals by a process of probability of mutation pmPair sub-population P2(t) performing selective differential mutation by individuals, wherein the selectivity is represented by: from P2(t) selecting an appropriate parent
Figure BDA0002192911830000081
According to
Figure BDA0002192911830000082
The value of index i is chosen among equations (3) through (5) in such a way as to perform differential variation to generate offspring individuals
Figure BDA0002192911830000083
Figure BDA0002192911830000084
Figure BDA0002192911830000085
Figure BDA0002192911830000086
Where i is 1,2, …, N/2, t is the number of iterations, pmIn this example, p is taken as the mutation probabilitymF is a scaling factor, taken in this example, of 0.2
Figure BDA0002192911830000087
To be slave population P2Of the different individuals selected at random,
Figure BDA0002192911830000088
it is a randomly selected non-dominant individual from an external archive set. When i% 3 is 1 (% is a remainder operation, the same applies below), the formula (3) is executed; when i% 3 is 2, performing equation (4); when i% 3 is 0, equation (5) is executed. Individuals in the formula (3)
Figure BDA0002192911830000089
Are all from P2(t) the individuals are randomly selected, so that the global search capability is strong, but the convergence speed is slow; equation (4) introduces non-dominant individuals in the external archive set
Figure BDA00021929118300000810
Promoting the generation of variant offspring individuals to distribute in
Figure BDA00021929118300000811
Nearby, the local optimization capability and inheritance of the algorithm are enhanced, the convergence speed is high, and the algorithm is easy to fall into local optimization; in formula (5)
Figure BDA00021929118300000812
And
Figure BDA00021929118300000813
is from P2(t) the randomly selected individuals of (t),
Figure BDA00021929118300000814
it is a non-dominant individual randomly selected from the external archive set, so that performing equation (5) can better keep the balance between global exploration and local optimization, but is less robust. The three differential variation strategies are respectively good and bad, and are used cooperatively to complement the advantages, so that the global exploration and local exploitation capabilities of the algorithm can be effectively balanced, and the convergence, diversity and robustness of the algorithm are improved.
Step 6: let T1(t)＝P1(t)∪P1'(t)，T2(t)＝P2(t)∪P2' (T), here T1(T) and T2(t) is a temporary intermediate population.
And 7: calculating T1(T) and T2(t) objective function value vectors for individual volumes.
And 8: respectively aiming at T according to the individual objective function value vectors calculated in the step 71(T) and T2(T) performing a fast non-supporting sort, utilizing T according to the sorting result1(t) the first N/2 superior individuals in the population update sub-population P1(T) use of T2(t) the first N/2 superior individuals in the population update sub-population P2(t)。
And step 9: by P1(t) and P2And (t) updating the external archive set by the non-dominated individual, and if the external archive set is full, executing a multiple maintenance strategy to maintain the external archive.
Further, both step 9 and step 4 involve grouping the sub-population P1(t) and P2(t) copying the non-dominant individual in (t) to the external archive set to update the external archive set. Due to the capacity limitations of the external archive set, once it appears that the external archive set is full, it is necessary to enforce multiple retention policies on the external archive set. The method of the invention adopts a collaborative epsilon-crowding distance method to maintain the diversity of the external archive sets, comprising the following steps:
(1) the initial value of epsilon parameter is set, in this embodiment, the initial value of epsilon is 0.1, and the size of external file set is set as N'.
(2) If the number of individuals in the external file set is less than N', the value of epsilon is reduced by the step length gamma, and the adjustment of the value of epsilon is stopped until the external file set is full. In this embodiment, the step length γ is 0.001.
(3) If a new individual is to be added into the external archive set, firstly, judging an epsilon-Pareto domination relation between the new individual and each individual in the external archive set; if the new individual is not dominated by any individual epsilon Pareto in the external archive set and the external archive set is not full, the new individual is added to the archive, and then epsilon Pareto domination relationships between all individuals in the external archive set are detected, and if dominated individuals appear, these dominated individuals are discarded. If the new individual and all individuals in the external archive set are in an epsilon-Pareto non-dominated relationship and the archive is full, calculating epsilon-congestion distances of the new individual and all individuals in the archive set, and abandoning the individual with the minimum congestion distance; if the new individual is dominated by the individual epsilon-Pareto in the archive set, the new individual is discarded.
Step 10: two sub-populations P1(t) and P2(t) by substitution, i.e. with P1(t) all individuals update P2(t) with P2All individuals in (t) update P1(t) where P is1(t) and P2(t) performing a substitution to give a sub-population P1(t) and P2(t) performing simulated binary crossing and selective differential mutation alternately as evolution progresses, and for a certain evolutionary generation, P1(t) and P2(t) can only be selected from two kinds of variation operations, namely crossover and variation, and the aim is to promote the co-evolution of two sub-populations.
Step 11: updating an iteration counter: t is t + 1.
Step 12: judging whether T reaches TmaxIf not, go to step 3; otherwise, go to step 13.
Step 13: and outputting all the individuals in the external file set, and finishing the algorithm.
Table 1 shows the IGD mean values obtained by the four multi-objective evolutionary algorithms over 28 examples. In order to reduce the influence of random errors on the calculation results, in this embodiment, each algorithm is independently run for 30 times in each calculation example, and the average value of the IGD indexes obtained by each algorithm on each calculation example is calculated. The IGD index measures the distance between the true Pareto front to the approximate Pareto obtained by the algorithm. Since the true Pareto fronts of all the examples in this embodiment are known, by performing diversity sampling on the true Pareto fronts of the examples, and calculating the distances between the sample points and the approximate Pareto fronts, not only the convergence of the algorithm can be reflected, but also the diversity of the solution set obtained by the algorithm can be measured. Generally, the smaller the IGD index value, the better the convergence and diversity of the algorithm.
The test result of the embodiment shows that the method can obtain the optimal IGD mean value on 27 of the total 28 algorithms, which shows that the multi-objective evolutionary algorithm based on double population cooperation has obvious performance advantages in convergence and diversity compared with the classical multi-objective evolutionary algorithm in solving complex multi-objective optimization problems with different characteristics, and proves the feasibility and superiority of the method.
FIG. 2 is an approximate Pareto front obtained by the method of the present invention on a 2-target ZDT1 baseline function, FIG. 3 is an approximate Pareto front obtained by the method of the present invention on a 3-target DTLZ6 test problem, FIG. 4 is an approximate Pareto front obtained by the method of the present invention on a 2-target WFG2 test function, and FIG. 5 is an approximate Pareto front obtained by the method of the present invention on a 2-target UF4 test problem. By comparing the real Pareto frontiers of these benchmark test functions, it can be found that the method of the present invention has better convergence and diversity effects.
TABLE 1 IGD means values obtained by four Multi-target evolutionary algorithms
Figure BDA0002192911830000101
Figure BDA0002192911830000111
While embodiments of the invention have been disclosed above, it is not limited to the applications set forth in the description and the embodiments, which are fully applicable in a variety of fields of endeavor to which the invention pertains, and further modifications may readily be made by those skilled in the art, it being understood that the invention is not limited to the details shown and described herein without departing from the general concept defined by the appended claims and their equivalents.

Claims (5)

1. The multi-objective evolutionary algorithm based on double population cooperation is characterized by comprising the following steps of:
s1: setting a target number M and a maximum iteration number Tmaxdecision vector dimension N, population size N, external archive set size N', pseudo-binary cross-distribution index η, variation probability pmDifferential evolution scaling factor F;
s2: making iteration counter t equal to 0, utilizing random initialization method to produce initial sub population P whose size is N/2 in search space1(0) Generating an initial sub-population P of size N/2 in a search space by using a mixed horizontal orthogonal initialization method2(0)；
S3: computing a sub-population P1(t) and P2(t) objective function value vectors for each individual;
s4: using a fast non-dominated sorting method to respectively sub-population P according to the objective function value vector of each individual calculated in step S31(t) and P2(t) sorting, copying the non-dominant individuals in the two sub-populations to an external archive set according to a sorting result; if the external archive set is full, executing a diversity maintenance strategy to maintain the external archive set;
s5: pair sub-population P1(t) performing the simulated binary crossover operation to generate the offspring population P thereof1' (t) for the subgroup P2(t) performing a selective differential variation to generate a progeny population P'2(t)；
S6: let T1(t)＝P1(t)∪P1'(t)，T2(t)＝P2(t)∪P′2(T), here T1(T) and T2(t) is a temporary intermediate population;
s7: calculating T1(T) and T2(t) the objective function value of each individualAn amount;
s8: the vectors of the individual objective function values calculated in step S7 are respectively used for T1(T) and T2(T) performing a fast non-dominated sorting, using T according to the sorting result1(t) the first N/2 superior individuals in the population update sub-population P1(T) use of T2(t) the first N/2 superior individuals in the population update sub-population P2(t)；
S9: by P1(t) and P2(t) the non-dominated individual updates the external archive set, and if the external archive set is full, a diversified maintenance strategy is executed to maintain the external archive;
s10: exchanging two sub-populations P1(t) and P2All of (t);
s11: updating an iteration counter: t is t + 1;
s12: judging whether T reaches TmaxIf not, go to step S3, otherwise, go to step S13;
s13: and outputting all the individuals in the external file set, and finishing the algorithm.
2. The dual-population-cooperation-based multi-objective evolutionary algorithm of claim 1, wherein the initial sub-population P in S21(0) And P2(0) Can be respectively expressed as
Figure RE-FDA0002450258600000011
Wherein,
Figure RE-FDA0002450258600000012
and t is the iteration number.
3. The dual-population-cooperation-based multi-objective evolutionary algorithm of claim 1, wherein the sub-population P in S51(t) performing the simulated binary crossover to generate its offspring, specifically: from P1(t) randomly selecting two parents
Figure RE-FDA0002450258600000021
And
Figure RE-FDA0002450258600000022
wherein
Figure RE-FDA0002450258600000023
To pair
Figure RE-FDA0002450258600000024
And
Figure RE-FDA0002450258600000025
executing the simulated binary cross operation to generate two offspring individuals
Figure RE-FDA0002450258600000026
And
Figure RE-FDA0002450258600000027
wherein
Figure RE-FDA0002450258600000028
Figure RE-FDA0002450258600000029
While
Figure RE-FDA00024502586000000210
Wherein i is 1,2, … n;
Figure RE-FDA00024502586000000211
u is a random number between [0,1] subject to uniform distribution, and eta is a cross-distribution index.
4. The dual-population-cooperation-based multi-objective evolutionary algorithm of claim 1, wherein the sub-population P in S52(t) performing a selective differential mutation operation to generate offspring individuals, here by a mutation probability pmPair sub-population P2(t) inPerforming selective differential mutation, comprising: from P2(t) selecting an appropriate parent
Figure RE-FDA00024502586000000212
According to
Figure RE-FDA00024502586000000213
The value of the index i selects one of the following three differential variation modes to perform variation operation so as to generate offspring individuals
Figure RE-FDA00024502586000000214
Figure RE-FDA00024502586000000215
Where i is 1,2, …, N/2, t is the number of iterations, F is the scaling factor,
Figure RE-FDA00024502586000000216
to be slave population P2Of the different individuals selected at random,
Figure RE-FDA00024502586000000217
it is a randomly selected non-dominant individual from the external archive set.
5. The dual-population-cooperation-based multi-objective evolutionary algorithm of claim 1, wherein the executing of the multi-retention strategy on the external archive sets in S4 and S9 comprises:
(1) setting an initial value of an epsilon parameter, and setting the scale of an external file set as N';
(2) if the number of individuals in the external file set is less than N', reducing the value of epsilon by the amplitude of the step length gamma generation by generation until the value of epsilon is not changed when the external file set is full;
(3) if a new individual is to be added into the external archive set, firstly, judging an epsilon-Pareto domination relation between the new individual and each individual in the archive set; if the new individual is not dominated by any individual epsilon-Pareto in the external archive set and the external archive set is not full, adding the new individual to the archive, then detecting epsilon-Pareto domination relations among all individuals in the external archive set, and if dominated individuals appear, discarding the dominated individuals; if the new individual and all the individuals in the archive set are in an epsilon-Pareto non-dominated relationship and the archive is full, calculating epsilon-congestion distances of the new individual and all the individuals in the archive set, and discarding the individuals with the minimum congestion distance; if the new individual is dominated by the individual epsilon-Pareto in the archive set, the new individual is discarded.
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